Simulation and Control
Post-processing of equation models derived out of the bond graph model can be done using many available
software. However, certain common insight can be gained even before proceeding to numerical calculations.
The controllability and observability of a linear plant is one of the key areas in automatic control theory.
The structural controllability and observability of a system can be ascertained beforehand through inspection
of causal paths in the bond graph model, though eventually they may be in disagreement with the obtained results
at a parametric level for some specific values. The state equations derived may be symbolically processed to create routh arrays that
lead to conditions of instability at an symbolic level. Qualitative reasoning and Fault diagnosis from known
qualitative signal values in a plant can be done through inspection and graph traversal according to causality
and power directions without taking recourse to lengthy interpretations at the equation level.
In this section, the fundamental concepts involved in interpreting, transforming and solving of bond graph
models are discussed.
1. Conversion to Signal Flow Graphs
2. State Space Control Models
3. Structural Controllability and Observability
4. Fault Diagnosis
1. Conversion of Bond Graphs to Signal Flow Graphs
Bond graphs for linear systems can be easily converted to signal flow graphs (SFG) and transfer functions between inputs and outputs can be obtained for further analysis using linear control theory. The procedure for such conversion is laid out here.
Nodes of SFG (Junction Equality Laws)
A signal flow graph constitutes of nodes that are connected by directed lines called branches. Each branch has an associated gain. Unlike bond graphs, SFGs do not have any causal and power information. The branches simply specify traversal path and are associated with only one signal. Each bond in a bond graph thus is represented by two nodes in a SFG (effort and flow node). Activated bonds are represented by only one node (e.g. a flow activated bond will be represented by an effort node and an effort activated bond by only a flow node).
All bonds connected to a 1-Junction have the same flow and are thus represented by a single flow node. Any flow activated bond at a 1-Junction is dropped from the SFG. Similarly, all bonds connected to a 0-junction are represented by a single effort node and effort activated nodes at that junction are not considered. The nomenclature for a flow node representing flow in bonds of a 1-junction is fi,j,k.., where i,j,k.. are the bond numbers of bonds connected to that junction. Effort nodes for bonds at a 0-junction are similarly represented by ei,j,k...
All bonds connected to a 1-junction contribute separate effort nodes (viz, ei, ej .., where i, j, .. are bond numbers) leaving apart those which are effort activated and those which have been already represented in SFG. Similarly, All non-flow activated bonds connected to a 0-junction contribute separate flow nodes (viz, fi, fj .., where i, j, .. are bond numbers) except those which have been already represented in SFG.
Branches and Gains (Elemental Relations)
For I-element in integral causality, the equation is f = m-1ò e dt.
Taking Laplace transform of both sides, f(s) = e(s)/(m s) or f(s)/e(s) = 1/(m s).
Thus the gain associated with an integrally causalled I-element is 1/ms while the branch is directed from the effort to flow node (cause to effect). Similarly, for a differentially causalled I-element, the gain is ms and the branch is directed from the flow to effort node (cause to effect).Cause and effect relation for an integrally causalled C-element is e = K ò f dt.
Taking Laplace transform of both sides, e(s) = K e(s)/s or e(s)/f(s) = K/s.
Thus the gain associated with an integrally causalled C-element is K/s while the branch is directed from the flow to effort node (cause to effect). For a differentially causalled C-element, the gain is s/K and the branch is directed from the effort to flow node (cause to effect).The relationship between cause and effect for R-elements are not described by any differentiation or integration. Thus the gain term doesn't contain the Laplace variable s. The gain for R-element in resistive causality is e(s)/f(s) = R whereas; for conductive causality, it is f(s)/e(s) = 1/R.
Similar relationships can be established for two-ports TF and GY. The bond graph elements in different causal patterns and their corresponding signal flow graph representations are shown in the table below.
Bond Graph Signal Flow Graph 
Receptors (Junction Algebra)
1-Junction is a flow equalizing and effort sum junction. The strong bond for the 1-Junction is the bond that has causality away from the junction. This strong bond thus provides information of flow to the junction. The weak variables of 1-Junction are efforts. The effort equation for the 1-junction is written for the weak variables (efforts), where the effort in the strong bond is expressed as signed sum of effort in other bonds. This weak effort variable of the strong bond is called the receptor of the junction. For the 1-junction shown below, the junction algebra equation is e2=e1-e3. The signal flow graph representation is shown to the right.
Bond number 2 is the strong bond and the weak variable e2 is the receptor node. Signals from other nodes for weak variables are added to this node. The gain for all branches in SFG corresponding to bonds that are power directed in opposite direction as compared to the strong bond (i.e. contra-oriented) is 1. Otherwise, for all co-oriented bonds (bonds having same power orientation as the strong bond as seen from the junction) the gain is -1.
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The receptor for a 0-junction is the weak flow variable of the strong bond ( the bond that decides effort of the 0-junction, i.e. causalled near the junction). For the 0-junction shown below, the junction algebra can be represented at the receptor node f2 as shown to the right.
Let us consider a system whose schematic diagram and bond graph model are shown below.
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In the first step, the left-most 1-Junction and elements connected to it are considered. Node e1 represents the input source. The part SFG is shown below.
The next junction considered for conversion is the 1-Junction for disk angular velocity connected with bonds 4,5 and 6. The resulting part SFG is shown below.
The gyrator between the two 1-junctions is then represented in SFG.
In the figure shown below, the next junction is the bond graph model (0-junction conected to bonds 7,8 and 9) is represented in the SFG.
The transformer representing rotational to linear velocity transform is then represented in the SFG.
The last 1-junction for rope dynamics (connected to bonds 9,10 and 11) is then modeled in SFG. The already existing flow node f9 is modified to depict flows of all the bonds at the 1-junction to which bond number 9 is connected.
Now the system has been completely represented as a signal flow graph. All available effort and flow nodes have been added. Signal from any flow node can be integrated (gain = 1/s) to represent a displacement node as shown in the figure below. Similarly, effort signal may be integrated to get the momentum node, flow signal may be differentiated (gain = s) to get the acceleration node, etc.
Let us now try to derive transfer function between the voltage source (node e1) and the lift of mass (node Q8). Lumping the loops in a way very similar to that used with block-diagrams can reduce the above SFG. The transfer function from input to output can be derived using Mason's gain rule discussed here.
G(s)= Si Pi Di / D ,
where Pi = gain of the i'th forward path,
D = 1 - S all individual loop gains
+S all possible gain products of two non-touching loops
-S all possible gain products of three non-touching loops
+ ...,
Di = the D for the part of the SFG which does not touch i'th forward path.
The following figure shows the forward paths and loops in the SFG for the motor-disk-load system.
The blue colored path from node e1 to node f9 is in the forward path, which then follows two paths one via e10 and other via e11 to the
terminating node Q8. Thus the number of forward paths is 2. The number of loops shown in the figure in different colors is 5.
L1 = Loop colored in red, gain = - m2 / (R J s).
L2 = Loop colored in violet, gain = - r2Rr / Js.
L3 = Loop colored in yellow, gain = - r2Kr / Js2.
L4 = Loop colored in pink, gain = -Rr / ms.
L5 = Loop colored in green, gain = -Kr / ms2.
Forward path gains P1 = m r Rr/ (R J m s3) and P2= m r Kr/ (R J m s4).
Both forward paths touch all loops and hence D1 = D2 = 1.
Gain products of two non-touching loops are L1*L4 and L1*L5. There are no three or more non-touching loops.
Thus the numerator of the transfer function is (m r Rrs + m r Kr)/ (R J m s4).
The denominator is 1 + m2/RJs + r2Rr/Js + r2Kr/Js2 + Rr/ms + Kr/ms2
+ m2/RJs * Rr/ms + m2/RJs * Kr/ms2.
Multiplying both numerator and denominator by RJms4, the transfer function obtained is
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m r Rr s + m r Kr RJms4 + (m2 m + r2RrR m + RrR J)s3 + (r2KrR m + KrR J + m2Rr)s2 + m2Krs |
Once the transfer function is obtained with symbolic coefficients, one may apply routh's criteria to obtain the stability conditions for both the open and closed loop system. Symbolic algebra can be performed manually or using software like Reduce, Mathmetica, etc. Assigning parameter values reduces the transfer function to ratio of two polynomials in 's' with numeric coefficients and usual control theoretical approaches can be used both in frequency domain (Bode, Nyquist plots etc.) and time domain (through inverse Laplace transform). Root loci analysis for stability analysis may be conducted. The transfer function can be converted to digital domain by taking z-transforms and then digital control system analysis can be performed.
More details on using bond graphs and signal flow graphs to design feedback control systems is available in the book "Modeling and Simulation of Engineering Systems Through Bondgraphs".
2. Control System State-Space Models
State-Space models are automatic output of the equations derived from a bond graph model after restructuring. The state-space
description of a dynamic system is given by the equations
d{X}/dt = A{X} + B{u},
{Y} = C{X} + D{u},
where {X} is the vector of states (P's and Q's), n is the number of states, A is a n x n square matrix, B is a n x m matrix, where
m is the number of sources, {u} is the vector of sources (SE's and SF's), {Y} is the vector of observer states (outputs), l is the number of
such outputs, C is lxn matrix and D is lxm matrix.
The transfer function matrix between input vector(s) {u} and output vector(s) {y} is given by
G(s) = C (sI - A)-1 B + D, where I is an nxn identity matrix.
Thus G(s) = C adjoint(sI - A) B / |sI - A| + D = n(s)/d(s),
where d(s) is matrix of the denominator polynomial and numerator polynomial n(s) = |sI - A|.
The poles of the system are thus eigen values of matrix A.
Symbolically deriving transfer functions of higher order systems is resource and time intensive. However, if numeric parameter values are assigned, the transfer function matrix can be easily obtained from numeric A,B,C and D matrices. These matrices A,B,C and D (collectively called a quadruple) can also be used in advance control theoretic analysis like controllability test, obervability test, pole-placement, noise-rejection filter design, etc.
Consider the system and it's bond graph model used in the earlier example.
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The state equations for the model are given below.
DP8 = Rr*(r*P5/J - P8/m) + Kr*Q11
DP5 = m/R*(SE1 - m*P5/J) - r*(Rr*(r*P5/J-P8/m) + Kr*Q11)
DQ11 = r*P5/J - P8/m
The state vectors are P8, P5 and Q11.
Let us define only one the observed output as velocity of the hung-mass or variable f8. From equations, f8 = momentum / mass = P8/m. Thus the A, B, C and D matrices are
C = [1/m 0 0],
D = 0.
Thus,
The denominator polynomial is
|sI-A| = (s+Rr/m)*(s*(s+m2/Rj + r2Rr/J) + r2Kr/J)
- Rrr/J*(s*Rrr/m+rKr/m)
- Kr*(Rrr2/mJ - (s+m2/RJ+r2Rr/J)/m).
= s3+(m2/RJ + r2Rr/J + Rr/m) s2
+ (Kr/m + r2Kr/J + m2Rr/mRJ) s + Krm2/mRJ.
The term in the first row and second column of the adjoint matrix of sI-A is Kr/J + (Rrr/J)s. Other terms of the matrix are not relevant since
both matrices [C] and [B] are sparse. So the numerator polynomial is (1/m)(Kr/J + (Rrr/J)s)(m/R).
After multiplying both numerator and denominator by mRJ, the transfer function obtained is
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m r Rr s + m r Kr RJms3 + (m2 m + r2RrR m + RrR J)s2 + (r2KrR m + KrR J + m2Rr)s + m2Kr |
The above transfer function from input excitation to velocity of hung mass can be integrated (by multiplying 1/s) to obtain transfer function up to displacement of the mass. The integrated transfer function is the same as that obtained through signal flow graph method. However, state-space models can be used for advanced control theoretical analysis, whereas transfer function models from SFG can be used for frequency domain analysis. Transfer function models can be realized in many possible state-space forms by canonical transformations, such as the control canonical form and the observer canonical form.
3. Structural Controllability and Observability
Before designing a system for a particular controlled output, the choice of sensors, actuators and their position plays the vital role. Often the configuration may lead to an uncontrollable or unobservable paradigm and the system may need redesigning. However, from a bond graph model representing the true system along with its actuators and observers, these control strategies may be verified for feasibility based on the analysis of structural properties of the model. Structural properties relate to properties depending only on the model structure (junction structure) and the elements (components) it is composed of excluding the dependence on numerical values of parameters.
The process of analysis of structural properties on bond graph models is due to G. Dauphin-Tanguy, A. Rahmani and C.Sueur as
presented in the paper titled "Bond graph aided design of controlled systems" in Simulation Practice and Theory,
Vol. 7, No 5-6, (1999) 493-513.
Some examples based on this theory are presented here without going through the actual proofs.
Definitions
Consider a system represented by the following state-space model.
d{x}/dt = A{x} + B{u},
{y} = C {x} + D{u},
where {x} is the state vector, {u} is the control input vector and {y} is the observer vector.
If n is the number of states (i.e. A matrix is nxn) then the characteristic polynomial for the system
or denominator of all transfer functions from control input to observed outputs may be written as
D(s) = |sI-A| = sp(sq + aq-1sq-1 + .. + a1s + a0),
where, p+q = n and the coefficients aq-1..a0 are functions of system parameters. The number of structurally null modes in the system is 'p'.
However, the actual null modes in the system may be higher owing to parameter dependence, e.g. when a0=0, the
number of null modes is 'p+1'.
The purpose of the study presented here is to find out the number of structurally null modes 'p' and the structural rank of the system 'q'.
Since the term sp can be separated out as a factor from the |sI-A|, the sI-A matrix contains p rows with diagonal elements containing s and other terms in those rows cancel out during determinant calcultation. Thus for s=0, or sI-A =-A, these rows are linearly dependent functions of other rows. This implies, the matrix A can be brought to a form, where p rows will contain all elements equal to zero. Thus, the structural rank of matrix A is n-p=q.
To find out 'q', the system equations need not be derived and tested. They can be obtained using the following simple rules based on the bond graph model.
The order of a system (dimension of matrix A or number of states) is the number of integrally causalled storage elements (I and C), when a preferred integral causality is assigned to the model.
The structural rank of the model (rank of matrix A without parameter dependence) is the number (q)of storage elements (I and C) that can be brought to differential causality, when a preferred differential (also called derivative) causality is assigned to the bond graph model.
The number of structurally null modes is the number (p=n-q) of storage elements (I and C) that cannot be brought to differential causality, when a preferred differential/ derivative causality is assigned to the model.
Controllability
The numerical method to determine controllability of a system is to calculate the rank of the controllability matrix. The controllability matrix is defined as [B AB .... An-1B]. The rank of controllability matrix depends on numerical values of the parameters and hence cannot be considered as an measure of robustness of the control strategy. This method also does not identify the modes of the system which are not controllable. The alternative numerical method is through feedback input injection, where the shift in eigen values of matrix A-BF (F is a gain matrix of small random values) from eigen values of A is considered as a measure of controllability of particular modes (eigen values). This method too is parameter dependent and is not a robust measure of controllability.
The structural controllability based on bond graph structure is however robust and parameter independent. The conditions needed for a bond graph model to be structurally controllable are
Every integrally causalled storage element (I and C) in the bond graph model with preferred integral causality must have at least one causal path linking it to a control source (control SE or SF).
Every integrally causalled storage element (I and C) in the bond graph model with preferred integral causality can be differentially causalled (assigned derivative causality) when preferred differential causality is assigned on the bond graph model without violating junction causality norms. If some integrally causalled elements in preferred integral causality mode cannot be assigned differential causality in the preferred differential causality mode and dualisation of some or all control sources (control SE to control SF and vice versa) allows them to accept differential causality, then the condition holds.
The part(a) is the attainability condition, whereas the part (b) pertains to the structural rank. If condition (b) holds, then the rank of matrix A is n and the system is controllable with a single actuator whose position is determined by the attainability condition in part (a) and other design considerations. If (b) does not hold, then the rank of matrix A is q<n, then for the model to be controllable, p=n-q actuators are needed and their location has to be determined by attainability conditions in part(a) and their participation has be determined by control source dualisation described in part (b).
Observability
The numerical method to determine observability of a system is to calculate the rank of the observability matrix. The observability matrix (O) is defined as
The rank of observability matrix depends on numerical values of the parameters and hence cannot be considered as an measure of robustness of the full-state measurement strategy. This method also does not identify the modes of the system that are not observable. The alternative numerical method is through feedback output injection, where the shift in eigen values of matrix A-FB (F is a gain matrix of small random values) from eigen values of A is considered as a measure of observability of particular modes (eigen values). This method too is parameter dependent and is not a robust measure of observability.
The robust structural observability is based on the satisfaction of the following conditions.
Every integrally causalled storage element (I and C) in the bond graph model with preferred integral causality must have at least one causal path linking it to a observer (effort activated C or flow activated I element, sometimes called Df and De, respectively).
Every integrally causalled storage element (I and C) in the bond graph model with preferred integral causality can be differentially causalled (assigned derivative causality) when preferred differential causality is assigned on the bond graph model without violating junction causality norms. If some integrally causalled elements in preferred integral causality mode cannot be assigned differential causality in the preferred differential causality mode and dualisation of some or all observers along with activation (flow activated I to effort activated C and vice versa) allows them to accept differential causality, then the condition holds.
If the rank of matrix A is n, then only one suitably placed observer (determined by attainability condition in part (a)) is sufficient to guarantee full-state observability. If the rank of A is q, then p=n-q number of additional observers are needed to assure observability and the position of these sensors has to be determined by the attainability condition and design considerations.
Example
Let us consider a system shown below. The two massive spools can be moved by injection of compressible fluid at the control inlet. The resistance between the cylinder and spools is neglected.
The bond graph model of the above system is shown below for both preferred integral causality and preferred derivative causality. The causal path linking the source to one of the spool masses is shown in the integrally causalled model.
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| Preferred integral causality | Preferred differential causality |
From the preferred integral causality model, the number of states of the system is 3. However, the
preferred differential causality model shows only 2 storage elements could be assigned differential
causality and one storage element is left in integral causality, which cannot be brought to differential
causality even through control source dualisation (converting SF to SE would cause violation of causal
structure at the 0-junction). Thus the rank of the system is 2 and it is not controllable.
Let us observe the state space representation of the integrally causalled model. The states are
P1, P2 and Q3. The state equations are
DP1 = K3*Q3,
DP2 = K3*Q3,
DQ3 = SF4 - P1/m1 - P2/m2.
Thus the A matrix is
The rank of this matrix is 2, which tallies with the results obtained earlier.
Let us now consider damping in one of the spools. The bond graph model for the damped
system in preferred integral causality and preferred derivative causality are shown below.
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| Preferred integral causality | Preferred differential causality |
In this case, all the storage elements with integral causality in the preferred integral causality model could be brought to preferred differential causality. Thus the damped system is controllable.
Let us now consider a modified system as shown below, where the massive spools have been anchored to the inertial frame.
The bond graph model for this system in preferred integral causality and preferred differential causality are shown below.
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| Preferred integral causality | Preferred differential causality |
The preferred differential causality model contains one state that was integrally causalled in the preferred integral causality model and could not be brought to differential causality in preferred differential causality model. However, dualisation of the control input can bring it to differential causality as shown below.
Preferred differential causality with dualised control source
Thus, the anchored system is structurally controllable.
Most bond graph software assign default integral causality. Thus the second part of rule that
assures structural controllability can be modified as follows.
Every integrally causalled storage element (I and C) in the bond graph model can be assigned
integral causality even when it is dualised, when preferred integral causality is applied.
If dualisation of some or all control sources
(control SE to control SF and vice versa) allows the dualised storage elements to
accept integral causality, then the condition holds.
Similarly, the rule for structural observability can be posed in an alternative way.
Let us consider a system shown below. The actual bond graph model and modified model after dualisation of integrally causalled storage elements in the former are shown in preferred integral causality.
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| System | Bond graph in preferred integral causality | Bond graph with dualised storage elements in preferred integral causality |
From the model, the causal paths between all integrally causalled storage elements with the control source as well as with the observer can be established and thus the attainability condition for both structural controllability and observability holds. The dualised storage element model has all dualised elements in integral causality. It did not require dualisation of control source or observer. The above model is both structurally controllable and observable.
Bicausal bond graphs can be used to investigate the controllability of systems in terms of the properties of the bond graph
of the inverse system. To learn more about this refer to the paper
Gawthrop P.J., Ballance D.J. and Dauphin-Tanguy G.,
"Controllability Indiacators from Bond Graphs",
in Proc. ICBGM'99, Edited by Jose J. Granda and
F. E. Cellier, Simulation Series, Vol 31, No. 1, SCS Publication, ISBN 1-56555-155-9.
You may download a copy from here (postScript File).
4. Fault Diagnosis using Bond Graphs
Most of the early approaches for fault diagnosis and isolation are rule based. Such approaches use simple prediction rules to provide possible causes and possible faults in inputs or members (parameters) of a system. These methods suffer from incompleteness and inflexibility.
Recent methods of fault diagnosis are based underlying structures and behavior of a system. Models serve as knowledge representation of a large amount of structural, functional and behavioral information and their relationship. This knowledge representation is used to create complex cause-effect reasoning leading to construction of powerful and robust automatic diagnosis and isolation systems.
All fault diagnosis methods can be broadly classified into two types, quantitative and qualitative. The quantitative approach relies on advance information processing techniques such as state and parameter estimation and adaptive filtering. The qualitative approach makes use of causal analysis which links individual component malfunctions expressed in qualitative form with deviations in measured values. This approach can be used when precise numerical models are not available, especially in the design stage.
Qualitative reasoning using bond graphs can be conducted to
construct intelligent supervisory control systems.
Qualitative reasoning is used as the general reasoning strategy through artificial
intelligence and bond graphs are employed as the knowledge representation.
Fault diagnosis mechanism helps to localize and isolate system faults.
It also helps in refinement of diagnosis method. These methods are then combined
together through a management mechanism to construct a supervisory control system.
The fundamentals of fault diagnosis presented here is based on the following works.
INTELLIGENT SUPERVISORY CONTROL, "A Qualitative Bond Graph Reasoning Approach" by H Wang & D Linkens (Univ. Sheffield), World Scientific Series in Robotics and Intelligent Systems - Vol. 14, ISBN 981-02-2658-6.
Khoda T., Inoue K. and Asama H., "Computer Aided Failure Analysis Using System Bond Graphs", in Proc. ICBGM'01, Edited by Jose J. Granda and Genevieve Duaphin-Tanguy, Simulation Series, Vol 33, No. 1, SCS Publication, ISBN 1-56555-221-0.
Feenstra P.J., Mosterman P.J., etal., "Bond Graph Modeling Procedures for Fault Detection and Isoaltion of Complex Flow Processes", in Proc. ICBGM'01, Edited by Jose J. Granda and Genevieve Duaphin-Tanguy, Simulation Series, Vol 33, No. 1, SCS Publication, ISBN 1-56555-221-0.
Linkens and Wang proposed use of qualitative equations from bond graph models instead of using
the differential equations. The qualitative representations for passive elements are defined as
R-element: e(t)=R f(t)
C-element: f(t)=C (e(t)-e(t-1))
I-element: e(t)=1/m (f(t)-f(t-1))
and qualitative states are defined as
[+] or [1] : higher than normal, increase
[0] : normal, no change
[-] or [-1] : lower than normal, decrease
The qualitative equations thus derived assume the values of components to be always
positive. Those parameters whose values are not known
are removed from qualitative equations. When a abnormal deviation in signature of
a particular measured quantity is observed, its value is entered into appropriate equation.
Fault is assumed in each parameter of the system (both + and - faults) and the value of the
power variable associated with the parameter is explicitly calculated using qualitative
operators. Then the fault is propagated by replacing the qualitative value of
assumed fault in place of qualitative value of measured value in each of the equations. The parameter
for which all the equations match without any conflict is considered as the
actual cause of the fault.
An alternative scheme based on causal structure analysis is presented here. This method initially creates a table of antecedents and consequences as described below.
Each power variable, parameter or moduli are divided into two classes, namely antecedents and consequences. For any equation x=f(y,z), where f() is a function, x is consequence whereas y and z are antecedents. When causal structure of each element and junction in the graph is considered, parameters always appear in the antecedent (cause) class.
For example, let us consider the case of 1-junction. The strong bond (one which is causalled away from the 1-junction) decides the flow variable of the junction and is the antecedent. Flow in all other bonds connected to the junction are consequences. The 1-junction is also an effort some junction. The effort in the strong bond is signed sum of effort in other bonds. Thus effort in the strong bond is consequence and effort in all other bonds is antecedents.
The antecedent and consequence classes for all junction types and elements are given in the following table.
| Component | Antecedents | Consequences |
| 1/m or I, e | f |
| m or 1/I, f | e |
| K or 1/C, f | e |
| 1/K or C, e | f |
| R,f | e |
| 1/R,e | f |
| f1,e2,m | e1,f2 |
| f2,e1,1/m | e2,f1 |
| f1,f2 | |
| f1,f2,m | e1,e2 |
 | f2 | f1, f3 |
| e1, -e3 | e2 | |
 | e2 | e1, e3 |
| f1, -f3 | f2 | |
 | SE | e |
 | SF | f |
For each model, the table of antecedents and consequences are prepared first. In this table each power variable appears only once in the antecedent and once in the consequence classes, whereas all parameters appear as antecedents.
Determination of Initial Fault Set (Single Fault Hypothesis)
Let us start with a single fault hypothesis. The initial fault set is determined through traversal of causal path. The faulty variable is first located in the consequences list and assigned a qualitative value (+1, 0 or -1). The back propagation is done by assigning qualitative values to corresponding antecedents. For instance, if a variable e1 is below normal (-1) and its antecedents are e2, -e3; then e2 must be assigned qualitative value -1 (below normal) and e3 must be assigned qualitative value -(-1)=1 (or above normal). Then these newly assigned antecedents are treated as consequences and antecedents for them are searched. A tree like structure thus results, which terminates at boundaries where parameters are detected as antecedents or the qualitative value of a detected antecedent is in conflict with an earlier value (which gets higher precedence or rank). The following example illustrates this approach.
Let us start with the simplest possible system and its bond graph model shown below.
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The table of antecedents and consequences for this model is given below.
| Antecedents | Consequences | |
| >Parameter | >Power Variable | |
| SE1 | e1 | |
| 1/m2 | e2 | f2 |
| K3 | f3 | e3 |
| R4 | f4 | e4 |
| f2 | f1, f3, f4 | |
| e1, -e3, -e4 | e2 | |
Let the single observed fault be f2 (velocity of mass) is lower than normal. For oscillating loads, amplitude of f2 may be considered lower than normal. Thus f2 can be assigned a qualitative value -1. By propagating this fault, a fault tree can be constructed as shown below.
In the above tree qualitative values are denoted using '+' for higher and '-' for lower. The conflicting nodes are represented in red color. Thus the initial fault set detected is M2(+), K3(+), SE1 (-1) and R4(+). Since we are working with single fault hypothesis, among these faults any one parameter could be the cause of the fault. Now constraints may be added to the tree or suggestions for qualitative measurement of the most appropriate variable/signal can be made. Let us assume the input excitation is not faulty (qualitative value 0). Thus a conflict appears at node for SE1. We may also assume that in such a system the mass cannot change over a long time of operation and the spring may weaken or fail during operation but cannot get stiffened. Thus qualitatively these are represented as M2 (0) and K3(0 or -). These two lead to conflicts at respective nodes. Thus the only parameter left is R4(+). The cause of low f2(-) can thus be R4(+) which corresponds to jamming or seizure of the damper.
Four Stage Logic in Single Fault Hypothesis
So far we have considered all variables to be positive. In practice, such a assumption may not work out too well. Consider a rotating machinery designed to rotate at a constant speed in anticlockwise direction. If this is considered as the positive angular velocity and we observe a fault that the machine rotates clockwise (i.e. in negative direction), then our present hypothesis considers the value has lowered or qualitative value of the signal is -1. However, the actual magnitude of the velocity in clockwise direction may be higher than the rated speed in anticlockwise direction. So, the sign independent algorithm is not applicable to general class of systems where large fluctuations in system behavior on occurrence of a single fault is possible.
The qualitative description of variables is thus enhanced to a larger set
to accommodate their sign.
1. Positive High (PH)
2. Positive Low (PL)
3. Negative High (NH)
4. Negative Low (NL)
5. Positive Normal (PN)
6. Negative Normal (NN)
The qualitative algebra for negation and inversion for above 6 types a defined in the table below.
| Quality | Negative | Inverse |
| PH | NH | PL |
| PL | NL | PH |
| NH | PH | NL |
| NL | PL | NH |
| PN | NN | PN |
| NN | PN | NN |
Considering the spring-mass-damper system of the previous example and its table of antecedents and consequences, the fault tree up to second level can be constructed as shown below.
However, if we proceed along the tree after second level, we would encounter some contradictory results such as qualitative values of K3 and R4 should be NL. The four stage logic thus introduces a reverse argument to deal with such incompatibility.
Let us look at the equation level, which implies e2=e1-e3-e4. Suppose the summed up effect of
qualitative values of e1 and -e4 is y. so the
qualitative value of e2(x1) and e3(x2) are related as x1=y-x3, where the '-' sign is qualitative negation.
Suppose x1=PL, then x3 may be NL (with y as PL) or it may be PH (with y as PH). This subjective judgment
is deferred and all the branches of the tree are constructed. Those branches leading to contradictory
results are removed from the tree and not propagated further.
Thus the negation operator always results in two branches. PL to NL and PH, PH to NH and PL,
NH to PH and NL, and NL to PL and NH. The general rule of negation for quality XY where XÎ(P,N)
and YÎ(H,L) is X(!Y) and (!X)Y, where ! stands for negation.
The new fault tree can then be constructed as follows.
Multiple Fault Hypothesis
Any observed fault in a system may be caused by failure of one or more components. At times component failures are linked together. This situation is handled by postulating all possible signal combinations at nodes representing signal algebra. Under this hypothesis, when the nodal variable happens to be the consequence of the weak law of the junction (algebra at a junction), multiple sets of branches are considered. We start with the two-stage fault logic, i.e. all parameters and signals are assumed positive.
If the weak law of a junction is A=B+C, where A,B and C are power variables, and qualitative value of consequence A is high, the antecedents B and C may have qualitative values as described below.
| A | B | C |
| High (+) = | High (+) | High (+) |
| High (+) | Low (-) | |
| Low (-) | High (+) |
Similarly, for low A(-) defined by A=B+C, B and C are both qualitatively low (-).
For high A(+) defined by A=B-C, B should be high(+) and C low(-).
For low A(-) defined by A=B-C, B can be low (-) and C low(-) or High(+). Also, B and C
can both be High (+) and their difference can still be low.
The tree representation for A=B+C is shown in the figure below.
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| Single Fault Hypothesis | Multiple Fault Hypothesis |
Based on the above hypothesis, the fault tree for the spring-mass-damper system can be drawn in stages as shown below.
Step-1
Step-2
Final step
In the above fault tree, same variable with same qualitative state is represented by a single node. This reduces the breadth of the tree structure. For larger systems, it is difficult to construct fault-tree manually. Normally such trees are represented as linked lists in a computer program.
The four stage logic accommodating signal sign creates larger combination of nodes and the tree grows exponentially with increase in system complexity. Automated fault diagnosis programs normally prune the tree by accepting qualitative inputs for other suggested (most critical) signals and resolving the conflicts (by removing those branches). The tree ultimately converges to a critical parameter list depicted by their interdependence. For the above fault tree, this chart is shown below.
| Qualitative value of parameters | |||
| SE1 | m2 | K3 | R4 |
| 0 | + | 0 | 0 |
| - | 0 | - | - |
| - | 0 | - | + |
| - | 0 | + | + |
| + | 0 | + | + |
| - | 0 | + | - |
| + | 0 | - | + |
| + | 0 | + | - |
Qualitative Analysis Using Tree Graphs
As proposed by Kohda et.al., this analysis starts with development of a bond graph model using the physical analogy and derivation of system equation. Then it follows both qualitative and quantitative failure effect analysis.
Qualitative analysis evaluates the final effects (steady state) under assumed failure conditions.
Quantitative analysis evaluates the transient effect leading to the final effect by utilizing detailed characteristics and information on system components.
At the steady state condition, it is assumed that the rate of states is zero. Thus their fluctuations in qualitative terms remains 0 at steady state.
The default nodes of a tree graph are states of a system, external inputs and derivative of states. The relationship of the states and inputs with derivative of states as described by the state equations is represented using two additional types of nodes, namely algebra and function nodes, as shown below.
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| Algebra | Function |
Initially, assumed faults are included in the system model. The equations for the global bond graph model is derived and represented as the tree graph.
Let us consider the model of a electrical circuit shown below. This model is dual of the two-tank system discussed in the paper by Kohda et.al. The possible faults in this model are malfunctioning of resistors, inductors and the input. Additional faults like resistances (R12,R14) and leakage (SF13,SF15) from inductor are also modeled. The load side is represented as SF11 element. The activated inductor I16 is used to measure output potential (Vout).
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The state equations for this model are
e2=SE1-R5*(P2/M2-P7/M7)-R12*(P2/M2-SF13)
e7=R5*(P2/M2-P7/M7)-R10*(P7/M7-SF11)-R14*(P7/M7-SF15)
Qualitative equations are written in terms of functions. Function expressions like C1(Q1) return K1*Q1 for
linear springs. Thus the above equation may be written in qualitative function form as shown below. These
functions are often termed characteristic functions which define behavior of elements.
dX1/dt=U1-R1(I1(X1)-I2(X2))-RL1(I1(X1)-U3)
dX2/dt=R1(I1(X1)-I2(X2))-R2(I2(X2)-U2)-RL2(I2(X2)-U4),
where X1 and X2 are mapped to states P2 and P7, U1 to U4 are mapped to inputs SE1,SF11,SF13 and SF15 and
the functions correspond to various system parameters.
The tree graph that represents the above state equations is given below. The starting nodes are the states of the system and the inputs or externals (unknown environmental effects). The terminating nodes are the derivatives of the states.
Let us now assume a failure in resistance R1, which may be due to burn out. Then R1 becomes a very high value (air resistance). This is represented as R1+ qualitatively or denoted by a + sign within a box near the function node R1. The steady state conditions imply both dX1/dt and dX2/dt are zero (constant). Also at steady state, the externals are not changing. So all these nodes and those representing static parameters are qualitatively assigned 0. The fault is then propagated according to the qualitative equations. The qualitative deviations are shown within a circle near the nodes. The fault tree graph is shown in the figure below.
The fault propagation starts from fixed nodes. Since change qualitative change in dX1/dt is zero, the nodes connected to it also experience no qualitative change. That implies the output from function node R1 is unchanged or qualitatively 0. However, the magnitude of R1 has qualitatively increased. So the magnitude of input signal has decreased (-). if we start from qualitative change in dX2/dt =0, then we observe qualitative change of the output from function node I2 is 0. At the node depicting output from function node I1 - output from I2, it is evident that qualitative value of output from function node I1 has decreased (-) since the output from function node I2 has not changed. This implies a decrease in value of state variable X1 (P11 in the bond graph model).
The tree graph shows the output from node R2 (potential across terminals or Vout) remain unchanged whereas the inductive momentum X1 (I1*current) decreases. This is evident from analysis that an increased R1 increases the overall impedance of the circuit and reduces the current drawn from the source.
To confirm these results, numerical simulation results obtained using SYMBOLS Shakti are
presented here. The representative parameter values are taken initially are
V=250V
I1=0.01H
I2=0.02H
R1=1W
R2=1W
RL1=RL2=0W (no leakage)
During the simulation, the fault was introduced by changing the value of R1 to 10W after
the system almost reaches the steady state.
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The results show decrease in the value of state X1 and a temporary increase in the output voltage that returns to the earlier magnitude in the new steady state. The quantitative simulation results thus validate the results obtained for steady state conditions through qualitative fault tree analysis.
These transient trends in the response obtained through quantitative analysis can be used to take corrective measures via intelligent supervisory control schemes. Faults such as resistance developing in inductor of the previous model can be likewise isolated through output observations.
Fault Detection and Isolation using Temporal Causal Graphs
Temporal causal graphs can be derived from bond graph models using the procedure for creating signal flow graphs. The '1/s' terms in frequency domain representing Laplace transform of integration with respect to independent variable t (time) are represented as integrals in temporal causal graphs. The integration with respect to time in representing storage elements introduces delay in system response. This delay is depicted as 'dt' in temporal causal graph.
The temporal causal graph can be traversed in both forward and back ward direction from an observed fault in a signal or parameter. The backward propagation implicates and isolated faulty parameters, whereas the forward propagation derives predictions for behavior of hypothesized faults.
Let us take an example of the system discussed earlier in the signal flow graph section. The system and its bond graph model are shown below.
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The signal flow graph for this system shown earlier has been modified to a temporal causal graph as shown below.
Let us consider a fault in velocity of the mass being lifted. If the lifting velocity is below its nominal value, i.e. f8-, then backward propagation (along path opposite to signal flow direction) yields 'm' or load mass is higher (m+) or e7,8,9 is lower (e7,8,9-). Continuing back propagation yields other possible results such as Kr-,Rr- (damage to the rope) and f7-.
f7- leads to r - (decrease in pulley radius) and J+ (increase in rotary inertia of pulley). This way, the entire path can be traced back to the input source.
The forward propagation of the temporal causal graph yields predictions for future behavior for the set of measurements for a postulated fault. This prediction takes account of temporal delays encountered during forward propagation. Let us consider a hypothetical case of temporary collapse of motor field (m-). Forward propagation (along the direction of signal only) leads to e5-. At this node, temporal edge (dt in forward path) is encountered and therefore effect on subsequent nodes is time delayed. This implies derivative of f4 is affected instead of its magnitude. This is represented as f40,-, where numeral '0' in superscript is an zero based index for the number of delays encountered during forward propagation to that node. This way, we can reach up to f9,10,110,- and e7,8,91,-. Further forward propagation yields e61,- and f82,-. Next step in propagation encounters nodes already traversed and hence the forward propagation is terminated. If not terminated, the results like e51,+ and f9,10,112,+ conflict with earlier qualitative values for those signals.
Normally, forward propagation is terminated when predictions are available for a sufficiently higher order. These predictions (also called signatures) obtained from forward propagation are listed as a table and are compared with the observed measurements to reduce the set of hypothesized faults to the actual ones.
External fluctuations and envisaged faults can be added to the model to study their qualitative effect on the system. For the system under consideration, failures like field collapse, ambient resistance on the rotating pulley and load, increase in pulley radius due to continuous winding, etc., can be added to the model and a new temporal causal graph (TCG) can be created and studied to generate global fault signatures.
Further information and literature on Temporal Causal Graphs can be obtained from the following links.
Transcend : Diagonosis of Continuous Dynamic Systems from Transients
Building observers to address fault isolation and control problems in hybrid dynamic systems